From the GEORGIA INSTITUTE OF TECHNOLOGY
An international team of scientists has developed a relatively simple mathematical explanation for the rogue ocean waves that can develop seemingly out of nowhere to sink ships and overwhelm oil platforms with walls of water as much as 25 meters high.
The waves stem from a combination of constructive interference – a known phenomenon of colliding waves – and nonlinear effects specific to the complex dynamics of ocean waves. An improved understanding of how rogue waves originate could lead to improved techniques for identifying ocean areas likely to spawn them, allowing shipping companies to avoid dangerous seas.
Based on an analysis of three rogue waves observed at different oil platforms in the North Sea over the course of a decade, the research was scheduled to be reported June 21 in the journal Scientific Reports. The work was done by researchers at the Georgia Institute of Technology, University College Dublin, and the Institut FEMTO-ST CNRS-Université de Franche-Comté.
“We saw similar wave behaviors at all three oil platforms,” said Francesco Fedele, a professor in the Georgia Tech School of Civil and Environmental Engineering. “We found that the main mechanism responsible for generating these waves is the constructive interference of elementary waves enhanced by second-order bound nonlinearities.”
Rogue waves have been observed in oceans around the world. They typically last only 20 seconds or so before disappearing, and are different from tsunami waves that can travel great distances after being created by underwater earthquakes or landslides.
Earlier research had suggested a phenomenon known as “modulational instability” to explain the rogue waves. That theory had been demonstrated in laboratories, but didn’t adequately explain the complex three-dimensional waves that were being measured in the open ocean without boundaries to constrain them, Fedele said.
Though ocean waves have a predominant direction, in the open ocean, waveforms from other directions can arrive. In rare conditions, those waves arrive in an organized way or almost in phase, leading to an unusual case of constructive interference that can double the height of the resulting wave.
But this doubled height still cannot explain the size of the rogue waves observed in the North Sea – and elsewhere. That difference can be accounted for by the nonlinear nature of the waves, which are not sinusoidal – but instead have rounded troughs, along with sharp peaks that result from the water being pushed upward against the pull of gravity.
“You have to account for the nonlinearity of the ocean, which is manifested in the lack of symmetry between the crests and the troughs,” said Fedele, who also has an appointment in Georgia Tech’s School of Electrical and Computer Engineering. “These nonlinear effects can produce an enhancement of 15 to 20 percent in wave height, which adds onto the effects of constructive interference.”
Using advanced mathematical techniques, the researchers modeled how waves could combine in very unusual circumstances to produce the rogue waves measured in at three different oil platforms in 1995, 2007 and 2015. Their model’s predictions match the waves measured.
“We describe the complex energy flow of a wave field by what we call its directional spectra,” said Frédéric Dias, a professor at University College Dublin. “What we have shown is that by combining knowledge of this spectra and using mathematics that accounts for second-order nonlinearities, we can reproduce the measured rogue waves almost exactly.”
While ocean waves can differ from other waveforms, the research team gained important insights from the optical community and the study of how light waves interact.
“These are fascinating results,” said John Dudley, a professor at the Institut FEMTO-ST CNRS-Université de Franche-Comté. “Many of us have spent years studying the effects of nonlinearity in wave amplification, but it is essential as a scientist to keep an open mind. It is not for us to tell Nature how to work – we must follow where it leads us, even if it means changing our ideas.”
The research has been the basis for a new rogue wave model that could be used to identify ocean areas where nonlinear effects could give rise to the waves. That could give shipping companies and others as much as an hour’s warning to avoid those areas.
In the end, Fedele said, the production of the rogue wave is simply chance: the rare combination of waves in what turns out to be a bad place for ships or oil platforms. “It’s just a bad day at the ocean,” he added.
In future work, Fedele hopes to apply the model to optical waves.
“What we would like to do next is show that there are wave groups in the ocean and in optics that behave in the same way,” he said. “There is an underlying physical entity which is the wave group. We see a wave packet, a group of waves that is moving and grows in amplitude to reach a maximum before it decays.”
In addition to those already mentioned, the research team included graduate student Joseph Brennan and postdoctoral researcher Sonia Ponce de Leon.
###
CITATION: Francesco Fedele, et al., “Real world ocean rogue waves explained without the modulational instability,” (Scientific Reports, 2016). http://www.nature.com/articles/srep27715
I suppose someone had to work out the mathematics on this. Still, as a former surfer this seems obvious. Waiting for that fabled “seventh wave” is just a matter of waiting for the energy of the various waves from various directions to align at a particular beach. Does this surprise any of the sea-going folk among us?
“… is just a matter of waiting for the energy of the various waves from various directions to align at a particular beach.”
The Wedge, Newport Beach, CA (4:38 minute video) …
Ok, need some surfer dude help with this. It looks to my eye like different wave fronts converging to form the peaks.. Is this from crossing swells or some diffraction of the same wave front around a feature???? Remembering the book I read “The Wave” I think; It wasn’t as technical as I would have liked but it was fun reading how these guys chase the long period swells down the coast.
Taz, the answer to your question is, “yes, both”. However, note that some of the crests were higher than others. This indicates the harmonic convergence of wave-forms, no matter whether they come from the breakwaters on the left of the video or from the various long-waves piling up on the beach. Asybot adds the issue of the shifting form of the sand underneath the waves.
In this video I believe that the breakwaters have more impact than the long-waves, although the rise of the bottom to the beach is steep and has a definite impact. (The waves break very close to shore. Not ideal for surfing, though the boogie boarders are having fun.)
Anyway, that is my “surfer dude” perspective.
For a good explanation go to Wikipedia and type in “the wedge”.
A big day at The Wedge …
Trying to figure out what caused the awesome waves in the video of July 2009 at the Wedge. I don’t see any major storms in the Atlantic (though I can’t find the full record). This is much more dramatic than the previous video and, frankly, is very impressive for an east coast beach.
“… very impressive for an east coast beach.”
Very impressive for the west coast as well, and in this case the west coast is where Newport Beach, CA is located.
I stand corrected and yes, very remarkable for a west coat beach. And that would explain the 2009 waves. I’ve been to New Port Beach, CA. (but not to surf). Did my surfing in Hawaii.
For a better perspective on the wave dynamics at The Wedge here’s a bird’s-eye view …
http://www.theinertia.com/surf/a-birds-eye-view-of-southern-californias-most-violent-wave/
For an interesting read on rogue waves Wiki-up “rogue wave”. The article talk of all sorts of waves besides water waves.
A take-away from the Wikipedia article …
At any given moment there are about ten rogue waves in existence on the world’s oceans.
Which means if your ship is in the wrong place at the wrong time then you are screwed.
I’m not a surfer, so Seventh Wave has a different meaning for me.
[youtube https://www.youtube.com/watch?v=u7mGjSZpdpk&w=640&h=360%5D
This is not a fascinating result. The referred constructive interference of waves assumes that these waves do not affect each other (no photon – photon interaction, for example). This is not the case for the waves of sufficiently high amplitude on a liquid surface, or sufficiently high amplitude acoustic waves in gas, liquid or solid. It is sad to see this trivializing of science on conceptual model level and over complication of mathematical model component. Not fascinating… Trivial nonsense…
“This is not the case….” That was the whole point! The non-linear effects are what they’ve been able to model.
I would be astounded if light in a vaccuum would behave in any kind of nonlinear way. In some kind of dielectric, I suppose it could happen.
Then again… it is the surprising results that make for advances. If they aren’t grant mining and there is some actual “there” there. Which sadly in climastrology Era science is not guaranteed.
This is exactly my point: there is no similarity at all. EM waves in vacuum (or in matter if the amplitude is “small”) do not affect each other. In case of surface waves – they always produce effect on each other. These academics become of such a low intellectual level that it is just painful to watch.
They have so little general knowledge that they simply don’t have the material to construct a proper B.S. filter.
Things would improve a lot if they were forced to read, and understand, whole papers and even whole books. As it stands, they spend most of their time mining for snippets they can use as citations.
It is a truism that most people forget 90% of what they learned in university. Given the evidence, I would say that most PhD candidates have forgotten 90% of what they learned in first year undergrad. LOL, sob, [/rant]
Bob,
Don’t forget that light rays, etc., all are seen to “bend” around large objects with hefty
gravitational fields… either following the warped space near the body or actually being
“bent” by the gravitation.
What of a light ray that encounters one of the recently reported “gravitation waves”?
All seen to produce nonlinear results.
I’m not sure that’s true.
With EM radiation, nonlinearities always create things like harmonics or sidebands, etc. If light beams bending in a strong gravitational field were associated with nonlinearity, we would see a change in the spectrum of the light beam. Nobody has ever brought such evidence to my attention.
If someone actually captures light bending by gravity full stop, wake me up :p
Happens all of the time. Advanced pulsed lasers and other electro-optical systems usually depend upon non-linear optical phenomena. They get femto-second laser pulses for studying very high speed interactions by compressing a longer pulse via non-linear optics.
Anyone who understands the implications of ocean wave spectra knows that while the average of the highest 1/2 waves ‘characterises’ the sea state ( it is generally called the significant wave height) and for many design purposes a wave of 1.67 Hs is used and for others the 1/1000 wave is used being 2Hs in height. I am not sure about the 1/10,000 or 1/100,000 wave heights relative to Hs but I am sure they are big.
As for mechanism, ocean spectra are made up of waves of a wide range of heights and lengths which therefore travel at different speeds and statistically there must be moments of coincidence that creates ‘rogue’ waves. As for wave trains moving in different directions, its called a confused sea. It seems to have confused this lot.
These guys seem to have discovered that the wheel is not just round but its circular too.
If they had popped in to a school of oceanography or naval architecture they would have had it all explained to them or they could have just read any one of a number of texts written decades ago.
This sounds like that new branch of science called ‘science communications’ arranging a vehicle for some funding.
the fact there is a higher incidence of these waves in the north sea compared to elsewhere is telling. strong current flows coupled with a tide shifting from flood to ebb with the wind in a certain quarter added to the effect of low or high pressure at either end of that body of water all add up to create the conditions required for this phenomena.
there is a spot on arbroath cliffs where occasionally you can watch huge peaks appear and disappear on spring tides several miles offshore with the wind from the north east and the ebb tide moving into the slack water period.the abrupt change in depth around the bell rock reef likely also contributes to this by shortening the distance between waves and increasing the peak height .
when these peaks occur they are three times the height of the surrounding waves. i have witnessed this twice in twenty years so not an overly regular occurrence.
People seem to be missing that what these people are claiming is not a novel qualitative feel for things but a quantitative model that gets the wave shapes right.
The new bit has to do with the non-linear nature of the ocean and the fact that the waves are higher then mere constructive interference would otherwise suggest.
M Seward,
I was reviewing casualty reports at the (UK) Marine Accident Investigation Branch a few days ago, so now offer his [unhappily, fatal] accident report –
https://assets.publishing.service.gov.uk/media/547c6fe9e5274a429000005d/MaerskKithiraReport.pdf
See the top of page ’13’ [19 of 30] for Table 2 – Maximum wave height as a multiple of the significant wave height and the probability of occurrence.
That report is dated 2008, so there is some science since then.
But, in short: –
“The research indicates that somewhere between 1 in 1000 and 1 in 2000 waves
will be twice the significant height. Thus, a vessel encountering waves with a
modal period of about 10 seconds, might expect to encounter a wave of twice
the significant height every 2¾ – 5½ hours, with the possibility that a wave of
nearly 2.5 times the significant height being encountered once every 11.5 days.”
I would add that the only really HUGE wave I met was in the Forties field in the North Sea, in April 1984. I was on the ‘Iolair’, IMO 7816060, as Extra Second Mate, the ink on my Master’s certificate still dampish.
Slow moving storms, to the NE, N and NWxW all produced decent wavefields – maybe significant heights of 7-10 metres each.
When all three peaked together, they troughed together, too, so there was a hole in the ocean in front of it, which we duly slid down.
The wave – high, steep, but thinnish, front to back, broke green over the wheelhouse – some 75-80 feet [23-25 metres] above MSL [and the sea level then was v e r y Mean!].
As soon as the weather eased we had a confusion of Naval Architects fly out to see what we had done to their precious ship.
Nothing, in my view.
Mother Nature had conjured up some lumpy water. She does.
And, yes, the bottom of the accommodation block – 15 metres/fifty feet above the water normally, had been set up in places by 18 inches/half a metre.
We had about seven guys with Masters’ certificates trying to explain cumulative wave-trains to the Naval Arch folk.
I am sure that is not the only cause of big nasty short waves.
Our friend – bit chilly, above – describes other circumstances when they come big.
And nasty.
A quote from the then current edition of the Mariner’s Handbook [NP100] – an essential book for seafarers – from 2008 or so: –
“A well found ship properly handled is designed to withstand the longest
and highest waves she is likely to encounter as long as they retain their
original shape. But when waves are distorted by meeting shoal water,
a strong opposing tidal stream or current, or another wave system,
abnormal steep fronted waves must be expected. Abnormal waves
may occur anywhere in the world where appropriate conditions arise. In
places where waves are normally large, abnormal waves may be massive
and capable of wreaking severe structural damage on the largest of
ships, or even causing them to founder.”
The above is a bit lengthy – apologies, but people who go down to the sea need to be aware of the perils of the sea, which are manifold, and may be fatal.
Auto
The topic captured my attention. The ads also caught my attention. I suspect that those that care about this website should be clicking them. They generate $. This is good. click.
“Using advanced mathematical techniques, the researchers modeled how waves could combine in very unusual circumstances to produce the rogue waves measured in at three different oil platforms in 1995, 2007 and 2015. Their model’s predictions match the waves measured . . . ‘These are fascinating results,’ said John Dudley, a professor at the Institut FEMTO-ST CNRS-Université de Franche-Comté. ‘Many of us have spent years studying the effects of nonlinearity in wave amplification, but it is essential as a scientist to keep an open mind. It is not for us to tell Nature how to work – we must follow where it leads us, even if it means changing our ideas.'”
I’m afraid these researchers are in for a rough ride. First, they began with something that actually happened in nature and constructed a model to replicate it. Then Professor Dudley had the unmitigated gall to announce that if nature doesn’t behave according to the model then ideas (and, by extension, the models) need to be changed.
Georgia Tech will undoubtedly be getting a surprise visit from an attorney general demanding they release all their research records for the past fifty years, and accusing them of taking money from big oil/coal/business to enrich the faculty and academic staff.
“What we would like to do next is show that there are wave groups in the ocean and in optics that behave in the same way,” but water waves obey Airy Wave Theory and are non-linear, and light waves in a vacuum are very linear. The key to the non-linearity of ocean waves is the presence of transverse potential gradients near the interface. The first place to try to draw an analogy would be internally reflected light waves on the interior surface of a gradient-index optical fiber.
Back in the 70’s, a friend and I, while on a return trip from Martha’s Vineyard encountered what could be called a rogue wave as we approached Woods Hole. We were sailing a fairly small sailboat – he being the sailor, and me being just a novice. It came, seemingly out of the blue, hitting us pretty much dead-on, which is a good thing, because if it had broadsided us I believe we could have capsized. It soaked us, and rattled our nerves. Our passage through Woods Hole was a hair-raising one as well, and later on, with deep swells, my friend wanted to put the spinnaker up to speed our return trip, and I reluctantly took the tiller. We very nearly capsized when I didn’t correct enough. He went into the water, managing to hang on, while screaming at me to let go of the tiller. Young and stupid, we were fortunate to make it home at all.
Wow – they figured out that waves interact? Wait until they discover that current in the water makes a difference!
Give ’em a few minutes on a small boat in the Gulf Stream when a good nor’easter is blowing. The square waves they’ll be bouncing around on might give them a better appreciation of mother nature… might also send ’em back to their models and a bit of redesign.
wind over tide and current, never a nice place to be, no matter what size the boat is.
It’s not clear to me what this research adds to the research referenced in https://wattsupwiththat.com/2009/04/05/the-1998-super-el-nino-possibly-a-rouge-wave/
E.g.
just from the press release, it sounds like the new thing here is that you have to take into account the non-sinusoidal shape of real waves to account for the full height of the rogue waves resulting from constructive interference.
Yes, the trochoidal shape of the waves is significant compared with the classical analysis and that is what this paper is about.
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/imgwav/trochoidpair.gif
But the shape of the wave varies tremendously depending on local factors – mostly current, but wind has an effect, too. Unless you know the exact state of local conditions, predicting wave shape is not going to be possible. I think they discovered this – the “second-order bound nonlinearities” that they describe – but they seem to think they can calculate these for analysis, which strikes me as a tall order in a non-lab environment.
And as any experienced mariner knows (ask WIllis!) it’s not just the high wave that gets ya, it’s the big hole between ’em.
I suspect that the wind and currents are what create what they are calling “wave spectra.”
@ hartley, 9:22 am, not only currents, wind etc but what a bout the sea floor topography I wonder if they take the undulations in the sea floor into account the whole equation changes again ( Surfers realize this as well)
The bottom would, of course affect wave shape, but here they were discussing waves that were “without boundaries to constrain them”, so I assume they were talking about waves in deep water, not where the bottom changes their behavior.
Is this a first year thesis in an Arts program?
I have the proof of wave interaction. I run my sea-doo around in tight circles, a large wave forms in the center. I then run through the center and jump that wave before it dissipates.
Can I get a reward for showing the proof?
I used to do something similar with a deep-v towing a float loaded with kids. Into the air they would all go.
Wave mechanics is interesting to me. Surface waves are of particular interest to me. I can see value in predictive modeling of odd-ball surface wave forms relative to ocean traffic and coastal structures….and just for the heck of it.
“but it is essential as a scientist to keep an open mind. It is not for us to tell Nature how to work – we must follow where it leads us, even if it means changing our ideas.” That is one of the most refreshing quotes I have seen from a scientist in some time.
100% with you Steve. Real science in action!
I also liked this bit: “Earlier research had suggested a phenomenon known as “modulational instability” to explain the rogue waves. That theory had been demonstrated in laboratories, but didn’t adequately explain the complex three-dimensional waves that were being measured in the open ocean without boundaries to constrain them, Fedele said.”
The theory worked in the lab, but not in real life. So they dug deeper. They didn’t make excuses or hide empirical evidence that didn’t agree with the lab results. They didn’t sit on their backsides and pretend to be all-knowing. They dug deeper. They did the science.
A joy to behold. Science lives! Just not in the loudest circles, it seems, but it’s out there. I enjoyed this article, it was indeed refreshing.
+1 , i am not quite getting the naysayers on this one.
Thanks, Bit Chilly. I don’t get the naysayers either. I can only think they didn’t pick up on the quality shown in this instance.
Chaos theory.
“it is essential as a scientist to keep an open mind. It is not for us to tell Nature how to work – we must follow where it leads us, even if it means changing our ideas.”
I think this is a great line. Has anyone explained this to the “climatologists”?
Interesting, but this explanation that it’s a cumulation of multiple waveforms has been around for a long time.
It’s pretty obvious that, as you put it, it’s a cumulation of multiple wave-forms. What appears to be new here is the way in which they see the cumulation occurring.
And in response to other comments along “seventh wave” lines, this is no seventh wave. They are talking about 3 waves in 3 different places over 20 years.
As others have pointed out, it’s very refreshing that they change theory to meet observation. Whether they have got it right this time remains to be seen, but we could certainly do with a bit more of that attitude in climate science.
But using sinusoidal waveforms which doesn’t lead to the observed magnitude of the ‘rogue’ waves. An ad hoc mechanism had been proposed to fix this (modulational instability), but didn’t explain the real world observations. This paper uses the actual shape of the oceanic waves (trochoidal) and found that constructive interference of these waves does lead to the level of amplification observed.
My grandfather who was a herring fisherman, talked about ‘a great lump of water’, something my father experienced when he was on duty on a UK weathership, somewhere in the North Atlantic in summer c.1954, in quite good weather. He’d been attending to his instruments in the Stevenson screen which was mounted on the bow of the old corvette, then went below decks to the lab. When he came back up to the bridge the skipper pointed forwards. The screen and instruments were gone, and the angle-iron that had supported everything was bent out of shape. Out of nowhere, ‘a great lump of water’ had come over the bow and swept the whole lot away. Luckily no-one was on the fore-deck at the time.
Your grandpa was lucky. I was caught on deck in the Great Australian Bight by one of these thinhgs on a 12,000 ton cargo ship when I was an Apprentice. In a way I was lucky because I survived although somewhat battered and bedraggled, and I have a hole in the back of my head to remind me of it.
(Who said I always talk out of a hole in the back of my head?)
A relation was sunk in the Solway by a smaller version some years later but all hands were rescued thank goodness.
the holes and sheer walls of water in the solway are a sight to behold on a bad day. i have run through the ross sound on the ebb tide with a fair bit of sou sou west running. watching daylight turn to darkness when the walls of water covered the boat and the aft deck full of water was interesting.
the slamming when dropping down the holes was deafening and tore the radar mount near clean off.
Roque waves, or Freak Waves have been studied for a good while, now. I don’t know that this work is an improvement over the work reported in the BBC documentary in 2002 here:
http://www.dailymotion.com/video/x225ztn_bbc-horizon-2002-freak-wave_shortfilms
That was a very interesting program. It recounted how, for many years, the “experts” maintained there were no such things as rogue waves, that they were simply the fanciful imaginings of seamen. Yet another example of how the scientific consensus can turn out to be completely wrong.
Rogue waves are easy to predict: turn your back on the ocean for about two minutes; turn around, and there will be a monster wave about to crush you into the sand. I relearn this lesson every summer’s vacation to Cape Hatteras.
On a serious note, my nephew and I experienced a rogue wave on the Patapsco River outside Baltimore, MD. We were coming out of Rock Creek heading ENE inside the White Rocks, and were about a half-mile from the south shore when I looked behind us to see that a four-foot wave was coming up on us from the SW. The average wave height was only about one foot, so this thing stood out like a mountain range moving across the water.
There were no boats between us and the south shore, so it wasn’t a wake, and it only had one crest anyway instead of a wake’s v-shape. There were no freighters anywhere in sight in the deep water channel ahead of us, so it couldn’t have been a rebound wave of a wake off the shore. Also the entire wave was only about 300 yards long and tailed off to nothing at the ends. It was also coming up on us fast.
It was unnerving to see that wall of water approaching our stern, but my boat (a 28′ sailboat) rose to it, gave us a bump as it passed under, and we watched it move away toward the channel, where it collapsed and disappeared. I still kick myself for not having a camera handy to get video of it.
Craziest I ever saw on the Bay in my six years of owning that boat.
There might be another explanation for some rogue waves; some rogue waves have been seen in the open ocean under relatively calm seas, which is not explained by the above.
One place is off the east coast of South Africa, where east travelling storm systems meet strong opposing west travelling ocean currents.
It has been suggested that when ocean currents and storm systems travel against and oppose each other, it may create a kind of wave front close to where the two systems meet, as waves and currents back up against each other, which lasts long after the storm has dissipated, creating rogue waves which continue to travel through the ocean in relatively calm seas.
There’s relevant local research and report (SA J Science, ca 1980; sorry, no ref).
The area is off South Africa’s Wild Coast (!). From memory: Scientists sailed a small research vessel into “Waratah weather” (after its most famous 1910 victim), and collected data while stopped in the water. They measured a 10m wave, and deduced it wasn’t a “rogue”; 20-30m waves are possible.
Their recipe for such waves was:
A strong deep-ocean swell from a storm E of the Drake Strait (set NE);
a local gale (“coastal low”) with wind SW and track NE;
increased Mocambique current (set SW);
the whole occurring along a narrow continental shelf with sharp drop-off.
IIRC they described a marked asymmetry of waveform – a steeper leading face, forming a “step in the ocean”. There’s anecdote in Mostert’s “Supership” of a laden supertanker in these conditions having a wave roll green the length of its deck.
Ah, yes. Noel Mostert’s “Supership” – great read! I believe he also tells of the HMS Queen Mary’s bridge getting swamped by a rouge wave on the North Atlantic while it was a troop ship after WWII. Now that’s a big wave!
The interesting thing is not that waves interact to produce a rogue wave but that the model identifies areas that produce such waves and warn of their possible presence.
As I recall there was an article several years ago specifically pointing out that the areas with mysterious shipping accidents, corresponded with areas subject to rogue wave action. I wish I’d kept the article.
Personal Experience ==> Several years ago, sailing from the northern Caribbean to Florida, we experienced a rogue wave (albeit, on the order of 10 feet in a sea of 3 foot swells). My son was at the helm, and my wife and I were each in one of the two forward cabins — asleep directly under the cabin top hatches, which were open to the breeze. We were rudely awakened by sea water pouring down as if the boat had been submerged.
My son explained that the normal swell was a point or two off the starboard bow, 2 to 3 feet, and all was well, when he looked up to see a 10 foot swell just yards off the bow — it broke over the bow, the blue water crashing down on the cabin tops — and in through the open forward hatches.
Just that one wave — in an otherwise normal and calm sea.
When a wave’s height exceeds its wavelength divided by 7, the wave exceeds a critical steepness and breaks to form a whitecap. In the diagram, the neighbouring waves were quite obviously wind-sea not swell and would therefore have been losing energy by white-capping as fast as they were being formed. The single big wave in the middle was much steeper than its neighbours. Why did it not break and form a whitecap?